An experimental demonstration is given of (nonlinear) iterative learning control applied to a reticle stage of a lithographic wafer scanner. To limit the presence of noise in the learned forces, a nonlinear amplitude-dependent learning gain is proposed. With this gain, high-amplitude signal contents is separated from low-amplitude noise, the former being compensated by the learning algorithm. Contrary to the underlying linear design, the continuously varying trade-off between high-gain convergence rates and low-gain noise transmission demonstrates a significant improvement of the nonlinear design in achieving performance.
In the past decades, the fabrication of integrated circuits has greatly benefited from improved lithographic technologies. Herein the control design of a reticle stage containing the patterns needed for illumination and a wafer stage containing the wafers to be illuminated is of major importance. In terms of feedback control, both reticle and wafer stages are controlled using PID-based control schemes on a single-input single-output basis. To obtain nano-scale position accuracy within less than milliseconds of settling time, the main part of the control effort, however, is induced by feedforward control.
In terms of feedforward, this paper considers iterative learning control (ILC) as a means to obtain zero settling times on a reticle stage in scanning direction; see, for example, Bien and Xu (1998), Moore and Xu (2000), Norrlöf (2000), and Xu and Tan (2003) for a thorough treatment of ILC, its design methods, and fields of usage. Roughly speaking, ILC refers to the iterative process of finding the learned commands needed to improve performance under repetitive motion using information from previous executions of such motion. In the lithographic field, the application of ILC is not new, see for example Rotariu, Ellenbroek, and Steinbuch (2003), Rotariu, Dijkstra, and Steinbuch (2004), Dijkstra (2004), but application on an industrial scale is not often seen. The main contribution of this paper, however, is the introduction of a nonlinear learning gain to continuously balance the trade-off between noise transmission/amplification and error convergence rates as a means to surpass linear control performance; all linear ILC techniques suffer to a certain extent from noise amplification—recurring disturbances are attenuated, nonrecurring are amplified; in this context, see Moore (1999) and Tayebi and Islam (2006). Under nonlinear learning, signal contents beyond a pre-defined threshold level is subjected to nonlinear weighting: larger signal levels correspond to larger learning gains. Below this level signals like small noises induce a zero learning gain and as such are excluded from the learning process.
Stability of the discrete-time nonlinear learning control is derived on the basis of Lyapunov theory, see also French and Rogers (2000) and Yakubovich, Leonov, and Gelig (2004), the latter for a recent contribution to this field. Herein a distinction is made between the nonperturbed case with no external inputs like noises and the perturbed case having such inputs. In both cases exponential convergence of the learning scheme is derived as long as the servo errors during subsequent iterations contain elements that exceed the pre-defined threshold level. Performance is expressed in terms of convergence as well as time-domain (settling) behavior. By adapting learning gains, rates of convergence are obtained at which the underlying linear learning schemes become unstable. In fact, nonlinear learning is shown to combine fast convergence with robust stability, see de Roover (1996), Gunnarsson and Norrlöf (2001), and Tousain and Van de Meché (2001) for linear approaches based on optimal control with a similar aim. Zero settling is demonstrated on an industrial reticle stage module. By itself, this significantly contributes in optimizing wafer throughput and, therefore, helps improving general performance of lithographic machinery.
The paper is organized as follows. First, the modelling, dynamics and control of a reticle stage are considered. Second, the ILC scheme is proposed including the introduction of the nonlinear gain filter and a motivation for nonlinear learning. Third, a Lyapunov-based stability and performance analysis is conducted with special focus on convergence and robustness properties. Fourth, an experimental demonstration in time-domain is given towards zero settling times on a reticle stage of an industrial wafer scanner. This paper is concluded with a summary of the main findings regarding nonlinear learning in the context of lithographic machinery
To improve performance of lithographic machinery in terms of wafer throughput, this paper aims at zero settling times under nonlinear learning control. Having a variable learning gain, the fixed trade-off between error convergence rates and noise transmission/amplification such as present in linear learning schemes is avoided. As a result fast convergence can be combined with limited noise amplification. This is because the error is quickly reduced (under high gain) towards a pre-defined threshold level. Below this level, signals are no longer subjected to learning (the gain becomes zero) and, therefore, can neither be transmitted nor amplified through the learned forces.
Stability of the nonlinear learning control is studied using Lyapunov analysis. With a single Lyapunov function candidate both the nonperturbed case (with no external inputs) as well as the perturbed case (with external inputs containing uniformly bounded noises) are shown to possess convergence properties. This relates to upper- and lower bounds on the number of trials needed to convergence to a fixed error level, but also to additional robustness to model uncertainty induced by the variable learning gains.
The nonlinear learning scheme demonstrates the ability of achieving zero settling times on an industrial short-stroke reticle stage module; the error signals in the settling region can no longer be distinguished from the remaining signals below the pre-defined threshold level. Learning therefore gives a significant improvement in performance as compared to the current PID-based feedback/feedforward control design. For both the learned forces and the servo error signals (nonmonotonic) convergence is demonstrated on a limited trial interval. The corresponding learning gain variation provides the basis for having a quick initial error reduction under large learning gains combined with sufficient robustness properties at further trials under smaller gains.
